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Testing strategies for monotonic and other qualitative trends
The psychological hypothesis to be examined may state that 'the
amount of retrieval (dependent variable) increases with increasing
values of imagery (the independent variable).' To test this
hypothesis J > 2 levels of imagery and an observable dependent
variable such as 'number of words correctly remembered'
are chosen. Omitting the psychological prediction referring to
the observable dependent variable and the design chosen, the statistical
prediction is derived from the hypothesis in an adequate and exhaustive
manner. This prediction refers to statistical concepts exclusively,
and since the discussion is restricted to (population) means the
resulting statistical prediction states a strictly monotonic trend
among the J means. This prediction has been called SPmon
in (9).
Since this statistical prediction cannot be tested appropriately
and exhaustively by a single test, it is then decomposed into
testable partial hypotheses about focussed pair contrasts. These
partial hypotheses can be tested in a way that enables unambiguous
(as far as test results are concerned) and testbased decisions
concerning the statistical prediction and that avoids any inconsistencies
stemming mainly from databased inferences.
'To avoid inconsistencies'
simply means: Ranks are only called 'different,' if
there are 'significant differences' among the means
according to the usual statistical criteria and tests applied,
whereby 'usual tests' refers to any twosample test,
whether it is a t test or a multiple comparison procedure
on pair contrasts, each with only one degree of freedom. Such
rankings are testbased.
The SPmon has already been presented
above, but is given here again:
The more experimental conditions that have been chosen, the greater the cumulation of error probabilities, but also the more severe the test of the psychological hypothesis, all other things being equal. The cumulation can be compensated for by an adequate adjustment, for example, by the DunnBonferroni method or an improved version of it (see, e.g., Kirk, 1982, pp. 106111 [36]; Westermann & Hager, 1986 [66]; Winer et al., 1991, pp. 158166 [68]). If the SHmon in (8) is connected with a lenient decision rule the acceptance of at least one partial alternative out of J1 partial hypotheses () suffices to accept the respective SPmon2 (). The decomposition is the same as before, but the decision rule is different. This means that the SPmon2 is more easily accepted than the SPmon1, but the test of the respective psychological hypothesis is less strict than with the SPmon1. An even more lenient decision rule gives leave to 'look for' the one necessary conforming result among all possible pairs of means which leads to the SPmon3. If tested according to this prediction the psychological hypothesis has an even less severe test to survive than if tested by the SPmon2.
In the derivation and testing of the SPmon2 or the SPmon3
the problem of possible rank inversions has not been discussed.
There are basically two options concerning rank inversions. First,
they are accepted if they occur when testing the SPmon2
or the SPmon3. Second, they are or at least a maximum
number of them is exluded a priori by a corresponding extension
of the decision rule. In this instance additional tests should
be planned referring to these inversions. Let us return to the
SPmon2 and extend its decision rule to handle possible
rank inversions; this extension leads to the SPmon4, which
deliberately allows for a maximum of rank
inversions a priori, suggesting the following decomposition:
The decision rule applied in the SPmon4 is more lenient than the one in the SPmon1, but stricter than the one in the SPmon3. Because of different numbers of pairs it is difficult to say whether the decision rule of the SPmon4 is stricter than the rule of the SPmon2, but since the SPmon2 allows for J2 rank inversions at most, the SPmon4 will most probably lead to a more severe test of the psychological hypothesis. Further decision rules or criteria concerning the maximum number of rank inversions and/or the number of pairs of means to be considered can be additionally defined, but will not discussed here. The recommendations are summarized in Table 2.
Next: Some further thoughts on Up: Qualitative Trends And Trend Previous: Some testing strategies aiming Methods of Psychological Research 1996, Vol.1, No.4 © 1997 Pabst Science Publishers 
